LTI Systems
The impulse response h[n] is the complete fingerprint of an LTI system. Know h[n] and you can predict the output for any input. This lesson unpacks FIR, IIR, difference equations, and how systems combine.
Apply δ[n] as input — the output is h[n]. That single measurement predicts the response to every possible input via convolution.
h[n] is nonzero for only M samples. The output is a finite weighted sum of M past/present inputs — no feedback required.
h[n] extends forever — implemented via recursive difference equations. Fewer coefficients for sharp frequency cuts, but stability must be checked.
h[n] = αⁿu[n] — stable when |α| < 1
Every digital filter is an LCCDE. The biquad (second-order section) is the standard building block — higher-order IIR designs are cascades of biquads.
Two LTI systems in series act as one system with combined impulse response h₁ * h₂. Order is interchangeable — h₁ then h₂ equals h₂ then h₁.
- Cascaded filters can be pre-combined offline into one kernel
- Commutative: swap order, same result
- Stable cascade requires each stage to be stable
Same input, outputs added — the combined system has h[n] = h₁[n] + h₂[n]. Used in graphic equalizers, multiband processors, and OFDM receivers.
- h[n] = T{δ[n]} — the impulse response fully characterizes any LTI system
- FIR: finite h[n], always stable, linear phase achievable, no feedback
- IIR: infinite h[n] via recursive LCCDE, fewer coefficients, check stability
- Biquad (2nd-order section) is the standard IIR building block
- Cascade: combined h = h₁ * h₂, order interchangeable
- Parallel: combined h = h₁ + h₂
- IIR stable ⟺ Σ|h[n]| < ∞ ⟺ all poles inside unit circle